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Internat. J. Math. & Math. Sci.

Vol. 8 No.4 (1985) 625-633

625

CONVEX CURVES OF BOUNDED TYPE

A. W. GOODMAN

Mathematics DepartmentUniversity of South Florida

Tampa, Florida 33620

(Received November 30, 1984)

ABSTRACT. Let C be a simple closed convex curve in the plane for which the radius of

curvature 0 is a continuous function of the arc length. Such a curve is called a convex

curve of bounded type, if 0 lies between two fixed positive bounds. Here we give a new

and simpler proof of Blaschke’s Rolling Theorem. We prove one new theorem and suggest

a number of open problems.

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. INTRODUCTION.

Let C be a simple closed convex curve in the plane. Such curves have been the

subject of numerous studies[l, 2, 3, 5, 6, 7, 8, 12, 14, 15, 19, and 24] to cite only a

few. Here we will refine the objects of study by looking at certain subsets. Through-

out this paper C is a simple closed convex curve in the plane for which the radius of

curvature 0 is a continuous function of arc length. Our refinement consists of putting

upper and lower bounds on O.

Definition i. We say that C is a convex curve of bounded type if there are con-

stants R and R2 such that

0 R 0 R2 (I.I)

at every point of C. We let CV(RI,R2) denote the set of all such curves that satisfy

(1.1) for fixed R and R2.Theorems about the class CV(RI,R2) appear in the literature (see for example

Theorem 3), but as far as I am aware, this class has not been given a specific name and

symbol until now. In this work we are concerned with one type of question, namely how

close can C come to its "center" and how far away from its "center" can C go.

The center can be defined in various ways. For example the center of mass of the

region bounded by C when the region has a uniform mass distribution. Or the center

could be the center of mass of the curve C when the mass is distributed either uniformly

or as some other function of s the arc length on C. In any case we can take the origin

as the center of mass without loss of generality. For each fixed curve in CV(RI,R2) set

D min IOPI, and D2 maxlOPl (1.2)PC PC

626 A.W. GOODMAN

Our main result is

Theorem I. Suppose that C CV(R I, R2) and the center is the center of mass of

the curve C. If the mass distribution on C is uniform, then

R =< I) _<_ D 2 =< R2 (1.3)

The two circles of radius R and R2 show that the inequality (1.3) is sharp.

Bose and Roy [6] call this center the perimeter centroid.

in section we review some facts about parallel curves and we give a new proof of

Blaschke’s Roiling Theorem [3, pp. 114-116]. In section 3 we prove Theorem 1. In

section 4 we suggest some topics for further research on the set GV(R1,R2).2. PARALLEL CURVES.

Let C e CV(R1,R2). We select the parameter s (arc length) so that s increases as

the point P P(s) traverses C in the counterclockwise direction. Let 0 denote, as

usual, the angle that the unit tangent T makes with the positive x-axis, and let N be

the unit inward normal to C at the point P P(s). Ue recall that

dx dycos 0, sin 0 (2 I)

ds ds

dxi dd-s _,Tds-

+ J- (cos 0)_i + (sin 0)j (2.2)

and

N (-sin 0)i + (cos 0)j. (2.3)

If V V(s) is the vector equation of C we introduce a second curve C* defined by

and

dy*= dy A< sin 0 (l-A<)sin 0. (2 6)ds ds

We let s*, *, and 0* denote, arc length, curvature, and radius of curvature at the

corresponding point on C*. Then (2.5) and (2.6) give

ds ] \ds ]+ (I-A) 2. (2.7)

if R A R2, then the curve C* may have cusps as shown in Fig. I. If A R

we set ds*/ds A 0. If A R2 then A 0 and we set ds*/ds

Thus in either case s* and s increase together. In the first case, A < R I, we havedV* d* dsT* [(I-A)cos 0 i + (l-A)sin 0 j] I-Adq* ds ds*

(cos )i + (sin )j T. (2.8)

If A R2, then the same type computation gives * -.Lemma I. If A R I, then the directed tangents at corresponding points of C and

C* are parallel and point in the same direction. Further N_* . If A R2, then T_*

-T and N* -N.

the vector equation V* V(s) + AN, where A is a constant. The curve C* is said to be

parallel to C, see [13 pp. 80-84, 18 p. 67, and 19]. Fig. shows a number of curves

parallel to the ellipse 2/9 + y2/4 I. The curve C* is also a Bertrand mate of C,

although the term Bertrand curve usually refers to twisted curves in space [4, p. 35].

If P(x,y) is a point on C and P*(x*,y*) is the corresponding point on the parallel

curve C*, then

x* x A sin and y* y + A cos 0. (2.4)

If I/0 is the curvature of C at P, then < d0/ds and from (2.4) and (2.1)

dx* dxds ds

A< cos (1-A)cos , (2.5)

CONVEX CURVES OF BOUNDED TYPE 627

A=O

=2

\

"’ + ’’,,

Figure

Lemma 2. If C CV(RI, R2) and A < RI, then Ca is locally convex and at corres-

ponding points 0 n 0 A.

By locally convex we mean *’ (s*) 0 at each point of Ca.Proof. From Lemma we have * at corresponding points. Hence, for the

curvature

, d* d d dsds* ds* ds ds* I-A/o"

Thus * 0 whenever O, and C* is locally convex. Further

(2.9)

0* I-A/00 (i A) 0 A (2 I0),

Of course 0 n 0 A is geometrically obvious from the definition of Ca. Q.E.D.

If A R2, the factor 1/(I-A/o) in (2.9) is replaced by 0/(A-0). Again Ca is

locally convex, but in this case 0* A 0.

It is geometrically obvious that if A R or A R2, then C* is a simple closed

curve. It seems that a direct proof is rather elusive. The difficulty may lie in the

following example. Let Ca be the image of Izl under the complex function f(z)

z + z 2. Then Ca is convex in the sense that * 0 at every point, so C* is locally

convex. But this curve fails to be simple. Nevertheless we have

Theorem 2. If C E CV(RI,R2) and A R or A R2, then C* is a simple closed

convex curve. If A RI, then Ca CV(RI*, R2*), where

RI* R A, and R2* R2 A. (2.11)

If A R2, then Ca g CV(RI* R2*), where

RI* A R2, and R2* A RI. (2.12)

Proof. We have already seen that C’is locally convex, but the example shows this

is not sufficient to prove that Ca is simple. On Ca letL*

A* *(e*) *(0) f ds*, (2.13)0 os

where L* is the length of Ca. We make a change of variables from s* to s. If A < R I,

then * and

d* dds ds"

Then (2.13) gives

L* L L d@ L, f d* ds ds* f d* ds f ds f d 27.0 d- d-* 0 d- 0 s 0

(2.14)

(2.15)

628 A.W.GOODMAN

Since C* is locally convex and A* 2, we see that C* is a simple curve.

If A R2, then * + . Hence (2.14) is still true and the proof remains valid.

The relations (2.11) and (2.12) follow from 0* 0 A and O* A 0 respectively. Q.E.D.

Theorem 3. Let C CV(R1,R2) and let K be a circle tangent internally to C at any

point PO of C. If K has radius RI, then K is contained in C. If K has radius R2, then

K contains C.

This theorem is often called Blaschke’s Rolling Theorem, because it states that

(a) a circle of radius R can roll around the inside of C, and (b) a circle of radius

R2 can roll around the outside of C. Blaschke has extended his theorem to 3-dimensional

space [3, p. 118]. For further work on this theorem, and various extensions see [II, 17,

20, and 22].

To be precise the phrase "internally tangent" means that K is tangent to C at POand the center of K lies on the inward normal to C at P0" Thus the location of the

center is given by equation set (2.4) with A replaced by R= the radius of the tangent

circle (a 1,2). We say that K is contained in C if K is contained in the closure of

the region bounded by C. Further K contains C, if C is in the closed disk bounded by K.

Proof of Theorem 3. We first show that the curve C cannot cross the circle K in a

neighborhood of P0’ the point of contact. Without loss of generality let PO be the

origin and let K and C be tangent to the x-axis at the origin. Further suppose that

both the circle and the curve lie above the x-axis, except at the origin. In this

position the lower half of the circle will have equation

Y R /R2_x2 -R x _< R. (2.16)

If y f(x) is the equation of C in a suitable neighborhood, I -e < x < e, then we

have f’(x) sgn x 0 and f’’(x) > 0 in I.

Lemma 3. Suppose that 0 R in I, where 0 is the radius of curvature on C. Then,

under the conditions described above

y(x) Y(x) R /R2 x2 x g I.

Thus in I, the curve C cannot cross from outside to inside K, but of course C may

coincide with K. We omit the proof of Lemma 3, but it follows directly from two inte-

grations, starting with the inequality

y’’(x)< (2.17)

3/2 R[l+(y’(x)) 2]

By reversing the inequality signs we have

Lemma 4. Suppose that 0 R in I. Then under the conditions on K and C des-

cribed above

y(x) J Y(x) R- /_X x e 1

Thus in I, the curve C cannot cross from inside to outside K, but of course C may

coincide with K.

From these two lemmas we see that if R R or R R2, then C cannot cross into or

out of K in a neighborhood of a point of tangency. To complete the proof of Theorem 3,

we must obtain this same result in the large.

First suppose that K has radius R and is tangent internally to C at PO" If K is

not contained in C, then K crosses C at a point P2 distinct from PO" Then we may find

a smaller circle K0 with radius R0 < R I, and such that K0 is tangent internally to C


at PO’ and is tangent to C at another point PI’ see Fig. 2.

Po

Figure 2

C*Now consider the parallel curve with A R0 < RI. By Theorem 2, this curve is

a simple close curve. On the other hand, the center D of the circle K0 is at least a

double point of C* because it is the corresponding point for both P0 and PI" Hence we

have a contradiction.

For the second part of Theorem 3 let K be a circle with radius R2 and tangent in-

ternally to C at P0" If K does not contain C, then K crosses C at a point P2 distinct

from PO" Then we may find a larger circle K0 with radius R0 > R2 and such that K0 is

tangent internally to C at P0 and is tangent to C at another point PI" Again consider

the parallel curve C* with A R0 > R2. By Theorem 2 this curve C* is a simple closed

curve. Just as before we obtain a contradiction because D the center of K0 is at least

a double point on C*. Q.E.D.

Corollary I. Let L(C) denote the length of C and let A(C) denote the area of the

region enclosed by C. If C E CV(RI,R2), then

2RI ! L(C) ! 2R2, (2.18)

and

RI2 ! A(C) ! R22" (2.19)

The circles of radius R and R2 show that both of these inequalities are sharp.

The inequalities (2.18) and (2.19) are well known, see for example [I, p. 352],

[15], and [16, Vol. I, pp. 529 and 548].

3. PROOF OF THEOREM I.

Let C g CV(RI,R2) and let (s) be a mass distribution of C. We exclude the trivial

case in which all of the mass is concentrated at one point. Then the center of mass will

be an interior point of the region bounded by C. Without loss of generality we select

the center of mass to be the origin. If L is the length of C, then

IL Lx(s)(s)ds 0 and y(s)(s)ds 0 (3.1)

0 0

630 A.W. GOODMAN

Now consider the parallel curve C* where A RI, and let * *(s*) be a mass distrl-

bution on C*. Then the moments Mx* and * are given by

* /L* x* (s*) * (s*) ds*0

and.L*

MX* 0 y*(s*)*(s*)ds*.

The change of variable from s* to s yelds

(3.2)

(3.3)

AIL (x- * (s*(s))(l-0-)ds, (3.4)M y* 0

andA

Mx* It (y+As) U* (s*(s))(l p--)ds. (3.5)

0

We now specialize, by setting (s) on C and selecting B*(s*) so that

u*(s*(s)) l-A/o(s) 0. (3.6)

Then (3.4) and (3.5) give

* IL (x-As)ds I

Lx ds A I dy 0,

0 0 C

and

Mx* (y+ ds y ds + A dx 00 C

from (3.1). Thus with the mass distribution (3.6), the center of mass of C* is also at

the origin. Since C* is a simple closed convex curve, the origin lies inside C* and

hence D A. Finally we note that A may be taken arbitrarily close to R E minlp for

points on C. Therefore

D RI. (3.7)

To prove that D2 R2, we consider the parallel curve C* where now A > R2. For

this curve equations (3.2)and (3.3) still hold. However, in this case we have

ds* A> 0. (3.8)ds

Thus in equations (3.4) and (3.5) we must replace the factor I- A/O by A/O I. If we

select (s) on C and u* on C A so that

(s*(s) o(s)A-o(s)

> O, (3.9)

then this mass distribution will give Mx* * 0. By Theorem 2, the curve C* is a

simple closed convex curve and the origin which is also the center of mass lles inside

the region bounded by C*.

No let P be a point on C furthest from the origin. Then OP is normal to C at P.If P* is the point on C* corresponding to P, then PP* is also normal to C at P. Hencethe points P, O, and P* are collinear.

Finally we observe that by Lemma I, the directed tangents to C and C* at the points

P and P* have opposite directions. Hence the origin is an interior point of the line

segment PP*. Therefore, IOP < IPP*I A. Since A may be taken arbitrarily close to

R2, we have D2 R2. Q.E.D.

4. FURTHER QUESTION FOR STUDY.

We observe that the inequality R D D2

R2

is sharp for the circles of

radius R and R2. But in each extreme case does not vary throughout the interval

[RI,R2] but instead is a constant at one end of the interval. The question naturally


arises, can we find better bounds for D and D2

p is a continuous function of s

whose values fill out the interval [RI,R2]. A first candidate for consideration is

the ellipse x a cos t, y b sin t, 0 _< t _< 27, with 0 < b < a. If we set

b (RI2R2)3, and a (RIR22) 3, (4.1)

then p fills out the interval [RI,R2]. Further D b and D2 a, so the expressions

in (4.1) may appear as the proper lower and upper bounds for D. If true, this would

improve the bounds R and R2 given in Theorem I. However, by piecing together arcs of

circles, we can show that no better bounds than R < D < D2 < R2 can be obtained. To

see this, we give C only in the first quadrant and complete the curve by reflecting C

in the x- and yaxes.

Let C and C2 be the two arcs defined by

x a + R cos t, x R2 cos t,

y RlSin t, y -b + R2 sin t, (4.2)

0 < t < T, T < t 7/2,

respectively. The endpoints of the two arcs will meet when t T if we select a

(R2-RI) cos T and b (R2-R I) sin T where 0 < R < R2. If we compute the first deriva-

tive for the two arcs at t T, they will not be equal, but the tangent vectors will be

parallel, so that for this choice of a and b, the curve C C u C2 is a smooth curve.

Further p R on C and 0 R2 on C2. Finally D R sin T + R2(l-sin T)and DI/R as

T/ 7/2. Similarly D2 R2 cos T + R (l-cos T) and D2R2 as T/0. Thus no better bounds

than D2 R2 and D R can be proved under the hypotheses stated. Of course p is no___t

continuous in a neighborhood of P(T), where C and C2 meet, but it is merely a matter of

labor to alter the curve slightly at P(T) to make 0 continuous.

Perhaps some better bounds for D and D2 can be obtained if we impose a further re-

striction that the average values of 0 over the curve be a fixed number such as (RI+R2)/2.One can also examine the problem of finding sharp bounds for D and D2 i the mass

distribution has some fixed pattern, other than uniform. For example, Steiner [23],and [24, pp. 99-159] has considered curves in which the mass distribution on C is pr6-

portional to the curvature at each point of C. More generally one can select the mass

distribution to be some other function of I/0.

One can also consider Theorem i, when the center of mass of C is replaced by the

center of mass of the region enclosed by C. With this replacement, Theorem was proved

earlier by Nikllborc [21] and Blaschke [2]. It is reasonably clear that the center of

mass of a curve C is in general different from the center of mass of the region en-

closed by C, but it may be of interest to examine a particular example.

Let C be f(Izl=l) under f(z) z + az 2, where 0 < a < I/4. Then C is symmetric

with respect to the x-axis and if the mass distribution is uniform on C then the center

of mass will be on the x-axis. Hence it suffices to compute the x-coordinate. Let

d and C denote this coordinate for the domain center of mass and the curve center of

mass respectively. As easy computation gives

d a(4 3)

i+2a2"

632 A.W. GOODMAN

A somewhat longer computation gives

MyRc --f-,

where

and

Hence

(4.4)

L f2/0 l+4a cos 0+4a2 dO, (4.5)

My /2(cos 0+a cos 20)/i+4a cos 0+4a2 dO (4 6)0

5 2+RC a(l- aIt is clear that in general d # C"

We may distinguish a third center of mass s, which we will call the conformal

strip center. Suppose that f(z) maps E conformally onto D, with f(0) O. Set Rs(r,l)the x-coordinate of the center of mass of the strip bounded by the curves

f(Izl=l) and f(IzI=r), where r I. Then by definition

s lim s(r,l). (4.7)r+l-

An easy computation shows that if f(z) z + az 2, 0 < a < I/4, and the mass distribu-

tion is uniform, then

s 2a

l+4a2 (4.8)

In this case s # d unless a 0. Further it is clear that in general s # C" This

example suggests the problem of finding

max lj-kl (4.9)

when C varies over the set CV(RI2) and j,k e {d,C,s}.

For other relations among various centers of mass, see Guggenheimer [I0], and Kubota

[19].

A computation, using Izl and

l’zf’(z)l (4 I0)P Re(l+zf"’(z)/’(z))’shows that for 0 < a < I/4, the function f(z) z + az 2 gives a convex curve

for which the radius of curvature is3/2

(I+4a cos B+4a2)l+6a cos 8+8a2 (4.11)

Extreme values of p occur when B 0, B , and cos B 2a. Thus z + az 2 is in

CV(RI,R2) for

R1 /l-4a2, and R2 (l-2a)2/(l-4a).One can also investigate the properties of normalized univalent functions that map

the unit disk conformally onto a region bounded by a curve in CV(RI,R2). Some elemen-

tary results in this direction have been obtained by the author [9].

REFERENCES

I. BALL, N. HANSEN On ovals, Amer. Math. Monthlv vol. 37 (1930) pp. 348-353.2. BLASCHKE, WILHELM ber die Schwerpunkte yon Eibereichen, Math. Zeit. vol. 36

(1933) p. 166.


3. BLASCHKE, WILHELM Kreis und Kugel, Walter deGruyter and Co., Berlin, 1956.

4. BLASCHKE, WILHELM Differential Geometrie vols. I and II, Chelsea Publishing Co.,New York, 1967.

5. BONNESEN, T. and FENCHEL, W. Theorie der Konvexen KSrper, Chelsea Publishing Co.,1948 New York, N.Y.

6. BOSE, R.C. and ROY, S.N. Some properties of the convex oval with reference to its

perimeter centroid, Bull. Calcutta Math Soc. Vol. 27 (1935) pp. 79-86.

7. CARATHODORY, C. Die Kurven mit beschrankten Biegungen, Sitz. der Preuss. Akad.Wiss. Phy. Math. Klasse (1933) pp. 102-125, Collected Works pp. 65-92.

8. FLANDERS, HARLEY A proof of Minkowski’s inequality for convex curves, Amer. Math.

Monthly vol. 75 (1968) pp. 581-593.

9. GOODMAN, A.W. Convex functions of bounded type Proc. Amer. Math. Soc., Vol. 92(1984) pp. 541-546.

I0. GUGGENHEIMER, H. Does there exist a "Four normals triangle"? Amer. Math. Monthlyvol. 77 (1970) pp. 177-179.

II. GUGGENHEIMER, H. On plane Minkowski geometry, Geometrica Dedicata vol. 12 (1982)pp. 371-381.

12. GUGGENHEIMER, H. Global and local convexity, Robert E. Krieger Publishing Co. Inc.1977 Huntington, N.Y.

13. GUGENHEIMER, H. Differential Geometry, Dover Publications 2nd edition 1977.

14. HAYASHI, T. On Steiner’s Curvature-Centroid, Science Reports Thoku Universityvol. 13 (1924) pp. 109-132.

15. HAYASHI, T. Some geometrical applications of Fourier series, Circolo Mat. Palermo,Rendiconti, vol. 50 (1926) pp. 96-102.

16. HURWITZ, A. Sur quelques applications gomtriques des s@ries de Fourier, Annalesde l’Ecole Normale suprieure vol. 19 (1902) pp. 357-408, Mathematische Werkevol. pp. 510-554.

17. KOUTROUFIOTIS, DIMITRI On Blaschke’s rolling theorems, Archiv der Math. vol. 23(1972) pp. 655-660.

18. KREYSZIG, ERWIN O. Differential Geometry, 1959 University of Toronto Press.

19. KUBOTA, TADAHIKO ber die Schwerpunkte der konvexen geschlossenen Kurven undFlchen, TShoku Math. Jour. vol. 14 (1918) pp. 20-27.

20. MUKHOPADHYAYA, S. Circles incident on an oval of undefined curvature, TShoku Math.Jour. vol. 34 (1931) pp. 115-129.

21. NIKLIBORC, WLADYSLAW ber die Lage des Schwerpunktes eines ebenen konvexen Bereichesund die Extrema des logarithmischen Flchenpotentials eines Konvexen Bereiches,Math. Zeit. vol. 36 (1933) pp. 161-165.

22. RAUCH, JEFFERY An inclusion theorem for ovaloids with comparable second fundamentalform, Journ. Diff. Geometry, vol. 9 (1974) pp. 501-505.

23. STEINER, JACOB Von dem Krummungsschwerpunkte ebener curve, Jour. reine angew. Mat.vol. 21 (1840) pp. 33-63 and 101-103.

24. STEINER, JACOB Gesammelte-Werke, Vol II Chelsea Publishing Co., New York, N.Y. 1971.

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